Asthma exacerbation or poorly controlled asthma: Given the patient's history of asthma and the use of an albuterol inhaler, it's likely that the cough is related to asthma, either as an exacerbation or due to poorly controlled symptoms. Asthma can cause chronic cough, especially if not well-managed.

Allergic rhinitis: This condition can cause postnasal drip, leading to a chronic cough. The absence of fever and the patient's age make this a plausible diagnosis, especially if the child is exposed to allergens.
Gastroesophageal reflux disease (GERD): GERD can cause a chronic cough in children, especially if stomach acid flows up into the throat, irritating the airways. This condition is more likely if the child experiences symptoms like heartburn or if the cough worsens after eating.
Upper respiratory tract infection (persistent or recurrent): Although the question mentions no fevers, upper respiratory infections can sometimes present without fever, especially in children with asthma, whose airways are more sensitive.

Pertussis (whooping cough): Although less common due to vaccination, pertussis can still occur, especially if the child's vaccination status is not up to date or if there's a waning immunity. It's crucial to consider this diagnosis due to its potential for severe complications and the need for public health intervention.
Foreign body aspiration: This is a critical diagnosis to consider, especially in children, as it can lead to severe respiratory complications if not promptly addressed. The absence of fever does not rule out this possibility.
Tuberculosis (TB): Although rare in many areas, TB is a critical diagnosis not to miss due to its implications for the child's health and public health. A chronic cough without fever could be an early sign, especially if the child has been exposed to someone with TB.

Cystic fibrosis: This genetic disorder can cause chronic respiratory symptoms, including cough, due to the production of thick, sticky mucus. It's less likely if the child has not had other symptoms like recurrent infections or failure to thrive, but it remains a consideration in the differential diagnosis of a chronic cough.
Habit cough: Also known as psychogenic cough, this is a rare condition where the cough is thought to be related to psychological factors rather than an organic cause. It's considered when other diagnoses have been ruled out and the cough seems to have a psychological component.
Bronchiectasis: A condition characterized by damage to the airways, leading to chronic cough and infections. It's less common in children without a history of severe infections or conditions like cystic fibrosis but should be considered if other diagnoses are ruled out.